Derived geometry of the first formal neighbourhood of a smooth analytic cycle

Grivaux, Julien

doi:10.1016/j.aim.2019.106924

Cited by 4 publications

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“…Therefore, the quantized HKR isomorphism of [7] is induced by the canonical isomorphism [1]). Outside the quantized world, one of the only known result is the calculation of the derived tensor product O X L ⊗ O S O X done in [8]. Unwrapping what it means, we see that the description of the truncations obtained as equalizers can be interpreted as the isomorphisms…”

Section: Square Zero Extensionsmentioning

confidence: 99%

“…Geometric setting and quantized cycles. The first setting is the one of [8]: S is a locally trivial first order thickening of a smooth k-scheme X by a locally free sheaf I. There is an associated exact sequence 0 → I → E → Ω 1 X → 0 where E = (Ω 1 S ) |X , whose extension class η in Ext 1 O X (Ω 1 X , I) determines entirely S. Hence the sheaf…”

Section: Square Zero Extensionsmentioning

confidence: 99%

“…This result gives a full understanding of the situation except that the homotopy-Lie algebroid structure is delicate to compute practically. On the other hand, the self-intersection of X into its first formal neighborhood is computed in [8]. Our goal is to provide among other things an alternative conceptual approach of the main result of [8] by developing the theory of free dg-Lie algebroids initiated in [11] in a systematic way.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the self-intersection of X into its first formal neighborhood is computed in [8]. Our goal is to provide among other things an alternative conceptual approach of the main result of [8] by developing the theory of free dg-Lie algebroids initiated in [11] in a systematic way. In particular, we describe explicitly their enveloping and jet algebras, which is new up to the author's knowledge.…”

Section: Introductionmentioning

confidence: 99%

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Derived intersections and free dg-Lie algebroids

Grivaux¹

2020

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We study free dg-Lie algebroids over arbitrary derived schemes, and compute their universal enveloping and jet algebras. We also introduce derived twisted connections, and relate them with lifts on twisted square zero extensions. This construction allows us to provide new conceptual approaches of existing results concerning the derived deformation theory of subschemes. Contents 1. Introduction 2. Recollections 2.1. DG-modules 2.2. Graded derivations and DG-Lie algebroids 2.3. Universal enveloping algebra of a dg-Lie algebroid 2.4. Coproduct and jets 2.5. Generalities on bimodules 3. Atiyah bimodules 3.1. Construction via anchored modules 3.2. Duality for Atiyah bimodules 3.3. Derived connections and the HKR class 4. Universal enveloping algebra of free Lie algebroids 4.1. Structure theorem 4.

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Section: Square Zero Extensionsmentioning

confidence: 99%

Section: Square Zero Extensionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Derived intersections and free dg-Lie algebroids

Grivaux¹

2020

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“…est un problème compliqué, certains éléments de réponse étant fournis dans les articles [1], [5] et [12] et [6] :…”

Section: Dans Le Casunclassified

Modèles géométriques attachés aux paires réductives

Grivaux

2019

Preprint

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JULIEN GRIVAUXRésumé. L'objet de cet article d'exposition est de présenter une introduction à l'article [6], et d'expliquer plus généralement comment le dictionnaire entre théorie de Lie et géométrie algébrique, développé dans [4] à la suite des travaux [14] et [15], peut être consolidé au sens où les énoncés géométriques ne sont plus seulement des analogues de résultats algébriques, mais en sont des extensions pour certains objets de Lie dans des catégories dérivées.

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Hodge theorem for the logarithmic de Rham complex via derived intersections

Hablicsek

2020

Res Math Sci

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In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in Arinkin et al. (Algebraic Geom 4:394–423, 2017) and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex.

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Derived geometry of the first formal neighbourhood of a smooth analytic cycle

Abstract: If X is a smooth scheme of characteristic zero or a complex analytic manifold, and S is a locally split infinitesimal thickening of X, we compute explicitly the derived self-intersection of X in S. ContentsDate: 15/11/2015.

Cited by 4 publications

References 28 publications

Derived intersections and free dg-Lie algebroids

Derived intersections and free dg-Lie algebroids

Modèles géométriques attachés aux paires réductives

Hodge theorem for the logarithmic de Rham complex via derived intersections

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